# Filtered category

In category theory, **filtered categories** generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of **cofiltered** category which will be recalled below.

## Filtered categories

A category is **filtered** when

- it is not empty,
- for every two objects and in there exists an object and two arrows and in ,
- for every two parallel arrows in , there exists an object and an arrow such that .

A **filtered colimit** is a colimit of a functor where is a filtered category.

## Cofiltered categories

A category is cofiltered if the opposite category is filtered. In detail, a category is cofiltered when

- it is not empty
- for every two objects and in there exists an object and two arrows and in ,
- for every two parallel arrows in , there exists an object and an arrow such that .

A **cofiltered limit** is a limit of a functor where is a cofiltered category.

## Ind-objects and pro-objects

Given a small category , a presheaf of sets that is a small filtered colimit of representable presheaves, is called an **ind-object** of the category . Ind-objects of a category form a full subcategory in the category of functors (presheaves) . The category of pro-objects in is the opposite of the category of ind-objects in the opposite category .

## κ-filtered categories

There is a variant of "filtered category" known as a "κ-filtered category", defined as follows. This begins with the following observation: the three conditions in the definition of filtered category above say respectively that there exists a cocone over any diagram in of the form , , or . The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for *any* finite diagram; in other words, a category is filtered (according to the above definition) if and only if there is a cocone over any *finite* diagram .

Extending this, given a regular cardinal κ, a category is defined to be κ-filtered if there is a cocone over every diagram in of cardinality smaller than κ. (A small diagram is of cardinality κ if the morphism set of its domain is of cardinality κ.)

A κ-filtered (co)limit is a (co)limit of a functor where is a κ-filtered category.

## References

- Artin, M., Grothendieck, A. and Verdier, J. L.
*Séminaire de Géométrie Algébrique du Bois Marie (SGA 4)*. Lecture Notes in Mathematics 269, Springer Verlag, 1972. Exposé I, 2.7. - Mac Lane, Saunders (1998),
*Categories for the Working Mathematician*(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2, section IX.1.